Problem: Simplify the following expression: $z = \dfrac{30a^2 - 27a}{12a^2 - 15a}$ You can assume $a \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $30a^2 - 27a = (2\cdot3\cdot5 \cdot a \cdot a) - (3\cdot3\cdot3 \cdot a)$ The denominator can be factored: $12a^2 - 15a = (2\cdot2\cdot3 \cdot a \cdot a) - (3\cdot5 \cdot a)$ The greatest common factor of all the terms is $3a$ Factoring out $3a$ gives us: $z = \dfrac{(3a)(10a - 9)}{(3a)(4a - 5)}$ Dividing both the numerator and denominator by $3a$ gives: $z = \dfrac{10a - 9}{4a - 5}$